Lieb's concavity theorem, matrix geometric means, and semidefinite optimization
Accepted version
Peer-reviewed
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Repository DOI
Change log
Authors
Fawzi, Hamza https://orcid.org/0000-0001-6026-4102
Saunderson, J
Abstract
A famous result of Lieb establishes that the map (A,B)↦tr[K^* A^{1−t}KB^{t}] is jointly concave in the pair (A,B) of positive definite matrices, where K is a fixed matrix and t∈[0,1]. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational t∈[0,1]. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.
Description
Keywords
Matrix convexity, Semidefinite optimization, Linear matrix inequalities, Lieb's concavity theorem, Matrix geometric means
Journal Title
Linear Algebra and Its Applications
Conference Name
Journal ISSN
0024-3795
1873-1856
1873-1856
Volume Title
513
Publisher
Elsevier
Publisher DOI
Sponsorship
Hamza Fawzi was supported in part by AFOSR FA9550-11-1-0305. James Saunderson was supported by NSF grant CCF-1409836.